G+Smo
24.08.0
Geometry + Simulation Modules
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A univariate B-spline basis.
T | coefficient type |
KnotVectorType | the type of knot vector to use |
Public Types | |
typedef gsBSplineTraits< 0, T > ::Basis | BoundaryBasisType |
Associated Boundary basis type. | |
typedef gsBSplineTraits< 1, T > ::Geometry | GeometryType |
Associated geometry type. | |
typedef Basis_t ** | iterator |
Iterators on coordinate bases. | |
typedef T | Scalar_t |
Coefficient type. | |
Public Member Functions | |
T | _activeLength () const |
Returns length of the ``active" part of the knot vector. | |
gsMatrix< index_t > | active (const gsMatrix< T > &u) const |
Returns the indices of active (nonzero) functions at points u, as a list of indices. More... | |
void | active_cwise (const gsMatrix< T > &u, gsVector< index_t, d > &low, gsVector< index_t, d > &upp) const |
void | active_into (const gsMatrix< T > &u, gsMatrix< index_t > &result) const |
Returns the indices of active basis functions at points u, as a list of indices, in result. A function is said to be active in a point if this point lies in the closure of the function's support. More... | |
gsMatrix< index_t > | allBoundary () const |
Returns the indices of the basis functions that are nonzero at the domain boundary. | |
void | anchor_into (index_t i, gsMatrix< T > &result) const |
Returns the anchors (greville points) of the basis. | |
gsMatrix< T > | anchors () const |
Returns the anchor points that represent the members of the basis. There is exactly one anchor point per basis function. More... | |
void | anchors_into (gsMatrix< T > &result) const |
Returns the anchors (greville points) of the basis. | |
const gsBasis< T > & | basis (const index_t k) const |
Helper which casts and returns the k-th piece of this function set as a gsBasis. | |
const_iterator | begin () const |
iterator | begin () |
int | borderKnotMult () const |
Returns the multiplicity of the first ``significant" knot (i.e., the m_p+1st). If it is different from the multiplicity of the corresponding knot at the end, returns zero. | |
gsMatrix< index_t > | boundary (boxSide const &s) const |
Returns the indices of the basis functions that are nonzero at the domain boundary as single-column-matrix. | |
BoundaryBasisType::uPtr | boundaryBasis (boxSide const &s) |
Gives back the boundary basis at boxSide s. | |
gsMatrix< index_t > | boundaryOffset (boxSide const &s, index_t offset) const |
bool | check () const |
Check the BSplineBasis for consistency. | |
uPtr | clone () |
Clone methode. Produceds a deep copy inside a uPtr. | |
gsMatrix< index_t > | coefSlice (short_t dir, index_t k) const |
Returns all the basis functions with tensor-numbering. More... | |
gsSparseMatrix< T > | collocationMatrix (gsMatrix< T > const &u) const |
Computes the collocation matrix w.r.t. points u. More... | |
Self_t & | component (short_t i) |
For a tensor product basis, return the 1-d basis for the i-th parameter component. | |
const Self_t & | component (short_t i) const |
For a tensor product basis, return the (const) 1-d basis for the i-th parameter component. | |
virtual uPtr | componentBasis (boxComponent b) const |
Returns the basis that corresponds to the component. | |
virtual uPtr | componentBasis_withIndices (boxComponent b, gsMatrix< index_t > &indices, bool noBoundary=true) const |
Returns the basis that corresponds to the component. More... | |
virtual void | compute (const gsMatrix< T > &in, gsFuncData< T > &out) const |
Computes function data. More... | |
void | connectivity (const gsMatrix< T > &nodes, gsMesh< T > &mesh) const |
virtual void | connectivityAtAnchors (gsMesh< T > &mesh) const |
virtual gsBasis::uPtr | create () const |
Create an empty basis of the derived type and return a pointer to it. | |
short_t | degree (short_t i) const |
Degree with respect to the i-th variable. If the basis is a tensor product of (piecewise) polynomial bases, then this function returns the polynomial degree of the i-th component. | |
void | degreeDecrease (short_t const &i=1, short_t const dir=-1) |
Lower the degree of the basis by the given amount, preserving knots multiplicity. | |
void | degreeElevate (short_t const &i=1, short_t const dir=-1) |
Elevate the degree of the basis and preserve the smoothness. | |
void | degreeIncrease (short_t const &i=1, short_t const dir=-1) |
Elevate the degree of the basis by the given amount, preserve knots multiplicity. | |
void | degreeReduce (short_t const &i=1, short_t const dir=-1) |
Reduce the degree of the basis by the given amount, preserve smoothness. | |
gsMatrix< T > | deriv (const gsMatrix< T > &u) const |
Evaluate the derivatives,. More... | |
gsMatrix< T > | deriv2 (const gsMatrix< T > &u) const |
Evaluates the second derivatives of active (i.e., non-zero) functions at points u. More... | |
void | deriv2_into (const gsMatrix< T > &u, gsMatrix< T > &result) const |
Evaluate the second derivatives of all active basis function at points u. More... | |
void | deriv2Single_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const |
Evaluate the (partial) derivatives of the i-th basis function at points u into result. | |
void | deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const |
Evaluates the first partial derivatives of the nonzero basis function. More... | |
void | derivSingle_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const |
Evaluates the (partial) derivatives of the i-th basis function at points u into result. More... | |
std::string | detail () const |
Return a string with detailed information on the basis. | |
gsDomain< T > * | domain () const |
short_t | domainDim () const |
Dimension of the (source) domain. More... | |
T | domainEnd () const |
Returns the ending value of the domain of the basis. | |
T | domainStart () const |
Returns the starting value of the domain of the basis. | |
size_t | elementIndex (const gsVector< T > &u) const |
Returns an index for the element which contains point u. | |
gsMatrix< T > | elementInSupportOf (index_t j) const |
Returns (the coordinates of) an element in the support of basis function j. | |
void | elementSupport_into (const index_t i, gsMatrix< index_t, 1, 2 > &result) const |
Returns span (element) indices of the beginning and end of the support of the i-th basis function. | |
void | elevateContinuity (int const &i=1) |
Elevate spline continuity at interior knots by i. | |
const_iterator | end () const |
iterator | end () |
void | enforceOuterKnotsPeriodic () |
Moves the knot vectors to enforce periodicity. | |
gsMatrix< T > | eval (const gsMatrix< T > &u) const |
Evaluate the function,. More... | |
virtual void | eval_into (const gsMatrix< T > &u, gsMatrix< T > &result) const |
Evaluates nonzero basis functions at point u into result. More... | |
virtual void | eval_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const |
Evaluate an element of the space given by coefs at points u. | |
std::vector< gsMatrix< T > > | evalAllDers (const gsMatrix< T > &u, int n, bool sameElement=false) const |
Evaluate all derivatives upto order n,. More... | |
virtual void | evalAllDers_into (const gsMatrix< T > &u, int n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const |
Evaluate the nonzero functions and their derivatives up to order n at points u into result. More... | |
virtual void | evalAllDersSingle_into (index_t i, const gsMatrix< T > &u, int n, gsMatrix< T > &result) const |
Evaluate the basis function i and its derivatives up to order n at points u into result. | |
virtual void | evalDerSingle_into (index_t i, const gsMatrix< T > &u, int n, gsMatrix< T > &result) const |
Evaluate the (partial) derivative(s) of order n the i-th basis function at points u into result. | |
virtual void | evalFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const |
Evaluate the function described by coefs at points u, i.e., evaluates a linear combination of coefs x BasisFunctions, into result. More... | |
virtual void | evalSingle_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const |
Evaluate the i-th basis function at points u into result. | |
void | expandCoefs (gsMatrix< T > &coefs) const |
Helper function for transforming periodic coefficients to full coefficients. | |
index_t | firstActive (T u) const |
Returns the index of the first active (ie. non-zero) basis function at point u Takes into account non-clamped knots. | |
gsBasisFun< T > | function (index_t i) const |
Returns the i-th basis function as a gsFunction. More... | |
void | getComponentsForSide (boxSide const &s, std::vector< Basis_t * > &rr) const |
Returns the components for a basis on the face s. | |
virtual T | getMaxCellLength () const |
Get the maximum mesh size, as expected for approximation error estimates. | |
virtual T | getMinCellLength () const |
Get the minimum mesh size, as expected for inverse inequalities. | |
gsBSplineBasis (KnotVectorType KV, const bool periodic=false) | |
Construct BSpline basis of a knot vector. | |
gsBSplineBasis (std::vector< KnotVectorType > KV) | |
Compatibility constructor with input an std::vector containing a single knotvector. | |
gsBSplineBasis (const T u0, const T u1, const unsigned interior, const int degree, const unsigned mult_interior=1, const bool periodic=false) | |
Construct a BSpline basis. More... | |
unsigned | index (unsigned i, unsigned j, unsigned k=0) const |
index_t | index (gsVector< index_t, d > const &v) const |
bool | indexOnBoundary (const gsVector< index_t, d > &ind) const |
Returns true iff the basis function with multi-index ind is on the boundary. | |
bool | indexOnBoundary (const index_t m) const |
Returns true iff the basis function indexed m is on the boundary. | |
bool | inDomain (T const &pp) const |
True iff the point pp is in the domain of the basis. | |
void | insertKnot (T knot, int mult=1) |
Inserts the knot knot in the underlying knot vector. | |
virtual gsGeometry< T >::uPtr | interpolateAtAnchors (gsMatrix< T > const &vals) const |
Applies interpolation of values pts using the anchors as parameter points. May be reimplemented in derived classes with more efficient algorithms. (by default uses interpolateData(pts,vals) | |
memory::unique_ptr< gsGeometry < T > > | interpolateData (gsMatrix< T > const &vals, gsMatrix< T > const &pts) const |
Applies interpolation given the parameter values pts and values vals. | |
gsGeometry< T >::uPtr | interpolateGrid (gsMatrix< T > const &vals, std::vector< gsMatrix< T > >const &grid) const |
bool | isActive (const index_t i, const gsVector< T > &u) const |
Returns true if there the point u with non-zero value or derivatives when evaluated at the basis function i. | |
bool | isClamped () const |
Checks, if both endknots have multiplicity m_p + 1. | |
bool | isPeriodic () const |
Tells, whether the basis is periodic. | |
virtual bool | isRational () const |
Returns false, since all bases that inherit from gsBasis are not rational. | |
const KnotVectorType & | knots (int i=0) const |
Returns the knot vector of the basis. | |
gsMatrix< T > | laplacian (const gsMatrix< T > &u) const |
Compute the Laplacian of all nonzero basis functions at points u. | |
gsBasis< T >::domainIter | makeDomainIterator () const |
Create a domain iterator for the computational mesh of this basis, that points to the first element of the domain. | |
gsBasis< T >::domainIter | makeDomainIterator (const boxSide &s) const |
Create a boundary domain iterator for the computational mesh this basis, that points to the first element on the boundary of the domain. | |
memory::unique_ptr< gsGeometry < T > > | makeGeometry (gsMatrix< T > coefs) const |
Create a gsGeometry of proper type for this basis with the given coefficient matrix. | |
virtual memory::unique_ptr < gsBasis< T > > | makeNonRational () const |
void | matchWith (const boundaryInterface &, const gsBasis< T > &, gsMatrix< index_t > &, gsMatrix< index_t > &, index_t) const |
Computes the indices of DoFs that match on the interface bi. The interface is assumed to be a common face between this patch and other, with an offset offset. The output is two lists of indices bndThis and bndOther, with indices that match one-to-one on the boundary bi. More... | |
short_t | maxDegree () const |
If the basis is of polynomial or piecewise polynomial type, then this function returns the maximum polynomial degree. | |
short_t | minDegree () const |
If the basis is of polynomial or piecewise polynomial type, then this function returns the minimum polynomial degree. | |
virtual index_t | nPieces () const |
Number of pieces in the domain of definition. | |
int | numCrossingFunctions () const |
Returns number of functions crossing the boundary of the knot vector. | |
size_t | numElements (boxSide const &s=0) const |
The number of elements on side s. | |
void | numElements_cwise (gsVector< unsigned > &result) const |
Returns the number of elements (component wise) | |
unsigned | order () const |
Returns the order of the B-spline basis. | |
gsMatrix< T > | perCoefs (const gsMatrix< T > &coefs) const |
Helper function for evaluation with periodic basis. More... | |
const gsBasis< T > & | piece (const index_t k) const |
Returns the piece(s) of the function(s) at subdomain k. | |
std::ostream & | print (std::ostream &os) const |
Prints the object as a string. | |
void | reduceContinuity (int const &i=1) |
Reduces spline continuity at interior knots by i. | |
virtual void | refine (gsMatrix< T > const &boxes, int refExt=0) |
Refine the basis on the area defined by the matrix boxes. More... | |
void | refine_h (short_t const &i=1) |
Uniform h-refinement (placing i new knots inside each knot-span. | |
void | refine_k (const TensorSelf_t &other, int const &i=1) |
Increases the degree without adjusting the smoothness at inner knots, except from the knot values in knots (constrained knots of initial geometry) More... | |
void | refine_p (short_t const &i=1) |
p-refinement (essentially degree elevation) | |
void | refine_withCoefs (gsMatrix< T > &coefs, const std::vector< T > &knots) |
Refine the basis by inserting the given knots and perform knot refinement for the given coefficient matrix. | |
void | refine_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, const std::vector< T > &knots) |
Refine the basis by inserting the given knots and produce a sparse matrix which maps coarse coefficient vectors to refined ones. | |
void | refineElements (std::vector< index_t > const &elements) |
Refinement function, with different sytax for different basis. More... | |
virtual void | refineElements_withCoefs (gsMatrix< T > &coefs, std::vector< index_t > const &boxes) |
Refine basis and geometry coefficients to levels. More... | |
void | removeKnot (T knot, int mult=1) |
Removes the knot knot in the underlying knot vector. | |
void | reverse () |
Reverse the basis. | |
void | setDegree (short_t const &i) |
Set the degree of the basis (either elevate or reduce) in order to have degree equal to i wrt to each variable. | |
void | setDegreePreservingMultiplicity (short_t const &i) |
Set the degree of the basis (either increase or decrecee) in order to have degree equal to i. | |
void | setPeriodic (bool flag=true) |
If flag is true, tries to convert the basis to periodic (succeeds only if the knot vector is suitable). | |
index_t | size () const |
size More... | |
index_t | size (short_t k) const |
The number of basis functions in the direction of the k-th parameter component. | |
template<int s> | |
void | size_cwise (gsVector< index_t, s > &result) const |
The number of basis functions in the direction of the k-th parameter component. | |
virtual const gsBasis & | source () const |
virtual gsBasis & | source () |
unsigned | stride (short_t dir) const |
Returns the stride for dimension dir. | |
void | stride_cwise (gsVector< index_t, d > &result) const |
Returns the strides for all dimensions. | |
gsMatrix< T > | support () const |
Returns (a bounding box for) the domain of the whole basis. More... | |
gsMatrix< T > | support (const index_t &i) const |
Returns (a bounding box for) the support of the i-th basis function. More... | |
gsMatrix< T > | supportInterval (index_t dir) const |
Returns an interval that contains the parameter values in direction dir. More... | |
virtual short_t | targetDim () const |
Dimension of the target space. More... | |
gsVector< index_t, d > | tensorIndex (const index_t &m) const |
Returns the tensor index of the basis function with global index m. | |
virtual gsBasis::uPtr | tensorize (const gsBasis &other) const |
Return a tensor basis of this and other. | |
short_t | totalDegree () const |
If the basis is of polynomial or piecewise polynomial type, then this function returns the total polynomial degree. | |
void | trimCoefs (gsMatrix< T > &coefs) const |
Helper function for transforming full coefficients to periodic coefficients. | |
int | trueSize () const |
Returns the size of the basis ignoring the bureaucratic way of turning the basis into periodic. | |
index_t | twin (index_t i) const |
Only meaningfull for periodic basis: For basis members that have a twin, this function returns the other twin index, otherwise it returns the same index as the argument. | |
void | uniformCoarsen (int numKnots=1) |
Coarsen the basis uniformly by removing groups of numKnots consecutive knots, each knot removed mul times. More... | |
virtual void | uniformCoarsen_withCoefs (gsMatrix< T > &coefs, int numKnots=1) |
Coarsen the basis uniformly. More... | |
void | uniformCoarsen_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, int numKnots=1) |
Coarsen the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones. More... | |
void | uniformRefine (int numKnots=1, int mul=1, int dir=-1) |
Refine the basis uniformly by inserting numKnots new knots with multiplicity mul on each knot span. | |
void | uniformRefine_withCoefs (gsMatrix< T > &coefs, int numKnots=1, int mul=1, int dir=-1) |
Refine the basis uniformly. More... | |
void | uniformRefine_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, int numKnots=1, int mul=1) |
Refine the basis uniformly. More... | |
virtual const gsMatrix< T > & | weights () const |
Only for compatibility reasons, with gsRationalBasis. It returns an empty matrix. | |
virtual gsMatrix< T > & | weights () |
Only for compatibility reasons, with gsRationalBasis. It returns an empty matrix. | |
Evaluation functions | |
gsVector< index_t > | numActive (const gsMatrix< T > &u) const |
Number of active basis functions at an arbitrary parameter value. More... | |
gsMatrix< T > | evalSingle (index_t i, const gsMatrix< T > &u) const |
Evaluate a single basis function i at points u. | |
gsMatrix< T > | derivSingle (index_t i, const gsMatrix< T > &u) const |
Evaluate a single basis function i derivative at points u. | |
gsMatrix< T > | deriv2Single (index_t i, const gsMatrix< T > &u) const |
Evaluate the second derivative of a single basis function i at points u. | |
virtual void | numActive_into (const gsMatrix< T > &u, gsVector< index_t > &result) const |
Returns the number of active (nonzero) basis functions at points u in result. | |
virtual void | activeCoefs_into (const gsVector< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const |
Returns the matrix result of active coefficients at points u, each row being one coefficient. The order of the rows is the same as active_into and eval_into functions. More... | |
Static Public Attributes | |
static const short_t | Dim |
Dimension of the parameter domain. | |
Protected Member Functions | |
void | _convertToPeriodic () |
Tries to convert the basis into periodic. | |
void | _stretchEndKnots () |
Adjusts endknots so that the knot vector can be made periodic. | |
Protected Attributes | |
KnotVectorType | m_knots |
Knot vector. | |
short_t | m_p |
Degree. | |
int | m_periodic |
Denotes whether the basis is periodic, ( 0 – non-periodic, >0 – number of ``crossing" functions) | |
Geometry evaluation functions | |
These functions evaluate not the individual basis functions of the basis, but a geometry object which is represented by a coefficient matrix w.r.t. this basis object. For the format of the coefficient matrix, see gsGeometry. These functions have default implementations which simply compute the basis function values and perform linear combination, but they may be overridden in derived classes if a higher-performance implementation is possible. | |
gsMatrix< T > | evalFunc (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const |
Evaluate the function described by coefs at points u. More... | |
gsMatrix< T > | derivFunc (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const |
Evaluate the derivatives of the function described by coefs at points u. More... | |
virtual void | derivFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const |
Evaluate the derivatives of the function described by coefs at points u. More... | |
virtual void | jacobianFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const |
Evaluate the Jacobian of the function described by coefs at points u. Jacobian matrices are stacked in blocks. | |
gsMatrix< T > | deriv2Func (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const |
Evaluates the second derivatives of the function described by coefs at points u. More... | |
virtual void | deriv2Func_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const |
Evaluates the second derivatives of the function described by coefs at points u. More... | |
virtual void | evalAllDersFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, const unsigned n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const |
Evaluates all derivatives up to order n of the function described by coefs at points u. More... | |
static void | linearCombination_into (const gsMatrix< T > &coefs, const gsMatrix< index_t > &actives, const gsMatrix< T > &values, gsMatrix< T > &result, bool sameElement=false) |
Computes the linear combination coefs * values( actives ) More... | |
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inline |
Construct a BSpline basis.
u0 | starting parameter |
u1 | end parameter parameter |
interior | number of interior knots |
degree | degree of the spline space |
mult_interior | multiplicity at the interior knots |
periodic | specifies if basis is periodic or not |
Returns the indices of active (nonzero) functions at points u, as a list of indices.
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inherited |
Returns a box with the coordinate-wise active functions
u | evaluation points |
low | lower left corner of the box |
upp | upper right corner of the box |
Returns the indices of active basis functions at points u, as a list of indices, in result. A function is said to be active in a point if this point lies in the closure of the function's support.
[in] | u | gsMatrix containing evaluation points. Each column represents one evaluation point. |
[out] | result | For every column i of u, a column containing the active basis functions at evaluation point u.col(i) |
Reimplemented from gsTensorBasis< d, T >.
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virtualinherited |
Returns the matrix result of active coefficients at points u, each row being one coefficient. The order of the rows is the same as active_into and eval_into functions.
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inlineinherited |
Returns the anchor points that represent the members of the basis. There is exactly one anchor point per basis function.
The exact definition of the anchor points depends on the particular basis. For instance, for a Bspline basis these are the Greville abscissae. In general, evaluating a function at the anchor points should provide enough information to interpolate that function using this basis.
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inlineinherited |
Get a const-iterator to the beginning of the bases vector
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inlineinherited |
Get an iterator to the beginning of the bases vector
Returns the indices of the basis functions that are nonzero at the domain boundary. If an offset is provided (the default is zero), it will return the indizes of the basis functions having this offset to the provided boxSide. Note that the offset cannot be bigger than the size of the basis in the direction orthogonal to boxSide.
Reimplemented from gsBasis< T >.
Returns all the basis functions with tensor-numbering.
k | in direction |
dir | Detailed explanation: |
Tensor-numbering in N-variate tensor-product basis means that each basis function is assigned an identifier (i_0, i_1, ..., i_{N-1}). This function returns indices of basis functions with i_dir = k and the returned indices are numbering of the basis functions in the basis (i.e., 0,1, ..., basis.size() ).
Bivariate tensor-product basis functions have tensor numbering (a,b). Calling dir=0, k=1 gives all functions with tensor-numbering (1,b). Calling dir=1, k=3 gives all functions with tensor-numbering (a,3).
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inlineinherited |
Computes the collocation matrix w.r.t. points u.
The collocation matrix is a sparse matrix with u.cols rows and size() columns. The entry (i,j) is the value of basis function j at evaluation point i.
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virtualinherited |
Returns the basis that corresponds to the component.
b | The component |
indices | The row vector where the indices are stored to |
noBoundary | If true, the transfer matrix does not include parts belonging to lower-order components (i.e., edges without corners or faces without corners and edges) |
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virtualinherited |
Computes function data.
This function evaluates the functions and their derivatives at the points in and writes them in the corresponding fields of out. Which field to write (and what to compute) is controlled by the out.flags (see also gsFuncData).
The input points in are expected to be compatible with the implementation/representation of the function, i.e. they should be points inside the domain of definitition of the function
[in] | in | |
[out] | out |
Reimplemented in gsGeometry< T >, and gsConstantFunction< T >.
Returns the connectivity structure of the basis The returned mesh has vertices the rows of matrix nodes
Reimplemented from gsTensorBasis< d, T >.
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inlinevirtualinherited |
Returns the connectivity structure of the basis The returned mesh has the anchor points as vertices
Evaluate the derivatives,.
Evaluates the second derivatives of active (i.e., non-zero) functions at points u.
See documentation for deriv2_into() (the one without input parameter coefs) for details.
[in] | u | Evaluation points in columns. |
Evaluate the second derivatives of all active basis function at points u.
Input parameter u is a gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.
result is a gsMatrix of size (K * d) x N, where
K is the number of active basis functions at the evaluation point.
Each column of result corresponds to a column of u. It contains the "pure" and the mixed derivatives for each active basis function, "above" each other.
Example (bivariate): Let \(B_i(x,y)\), d = 2 be bivariate basis functions, and let the functions with indices 3,4,7, and 8 (K = 4) be active at an evaluation point u. Then, the corresponding column of result represents:
\( ( \partial_{xx}\, B_3(u), \partial_{yy}\, B_3(u), \partial_{xy}\, B_3(u), \partial_{xx}\, B_4(u), \partial_{yy}\, B_4(u), \partial_{xy}\, B_4(u), \partial_{xx}\, B_7(u), ... , \partial_{xy}\, B_8(u) )^T \)
Example (trivariate): Let \(B_i(x,y,z)\), d = 3 be trivariate basis functions, and let the functions with indices 3,4,7, and 8 be active at an evaluation point u. Then, the corresponding column of result represents:
\(( \partial_{xx}\, B_3(u), \partial_{yy}\, B_3(u), \partial_{zz}\, B_3(u), \partial_{xy}\, B_3(u), \partial_{xz}\, B_3(u), \partial_{yz}\, B_3(u), \partial_{xx}\, B_4(u), ... , \partial_{yz}\, B_8(u) )^T \)
[in] | u | Evaluation points in columns (see above for format). |
[in,out] | result | For every column of u, a column containing the second derivatives as described above. |
See also deriv2() (the one without input parameter coefs).
Reimplemented from gsTensorBasis< d, T >.
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inlineinherited |
Evaluates the second derivatives of the function described by coefs at points u.
See documentation for deriv2_into() (the one with input parameter coefs) for details.
[in] | u | Evaluation points in columns. |
[in] | coefs | Coefficient matrix describing the geometry in this basis. |
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virtualinherited |
Evaluates the second derivatives of the function described by coefs at points u.
...i.e., evaluates a linear combination of coefs * (2nd derivatives of basis functions), into result.
Evaluation points u are given as gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.
The coefficients coefs are given as gsMatrix of size N x n, where
N is the number of points = number of basis functions and
n is the dimension of the physical domain.
Each row of coefs corresponds to the coordinates of one control point.
Let the function \( f: \mathbb R^3 \to \mathbb R^3\) be given by
\[ f = ( f_1, f_2, f_3)^T = \sum_{i=1}^N c_i B_i(x,y,z), \]
where \( B_i(x,y,z)\) are scalar basis functions and \(c_i\) are the corresponding (m-dimensional) control points. Then, for each column in u, the corresponding column in result represents
\[ ( \partial_{xx}\ f_1, \partial_{yy}\ f_1, \partial_{zz}\ f_1, \partial_{xy}\ f_1, \partial_{xz}\ f_1, \partial_{yz}\ f_1, \partial_{xx}\ f_2, \partial_{yy}\ f_2, \ldots , \partial_{xz}\ f_3, \partial_{yz}\ f_3)^T. \]
at the respective evaluation point.
[in] | u | Evaluation points in columns (see above for format). |
[in] | coefs | Coefficient matrix describing the geometry in this basis. |
[out] | result | For every column of u, a column containing the second derivatives at the respective point in the format described above. |
This function has a default implementation that may be overridden in derived classes for higher performance.
See also deriv2() (the one with input parameter coefs).
Evaluates the first partial derivatives of the nonzero basis function.
Let
d denote the dimension of the parameter domain.
K denote the number of active (i.e., non-zero) basis functions (see active_into()).
N denote the number of evaluation points.
The N evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.
The gsMatrix result contains the computed derivatives in the following form:
Column j of result corresponds to one evaluation point (specified by the j-th column of u). The column contains the gradients of all active functions "above" each other.
For example, for scalar basis functions \(B_i : (x,y,z)-> R\), a column represents
\((dx B_1, dy B_1, dz B_1, dx B_2, dy B_2, dz B_2, ... , dx B_n, dy B_N, dz B_N)^T\),
where the order the basis functions \(B_i\) is as returned by active() and active_into().
[in] | u | Evaluation points given as gsMatrix of size d x N. See above for details. |
[in,out] | result | gsMatrix of size (K*d) x N. See above for details. |
Reimplemented from gsTensorBasis< d, T >.
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inlineinherited |
Evaluate the derivatives of the function described by coefs at points u.
u | evaluation points as N column vectors |
coefs | coefficient matrix describing the geometry in this basis, n columns |
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virtualinherited |
Evaluate the derivatives of the function described by coefs at points u.
Evaluates a linear combination of coefs*BasisFunctionDerivatives, into result.
This function has a default implementation that may be overridden in derived classes for higher performance.
Let the function \(f: \mathbb R^d \to \mathbb R^m \) be described by the coefficients coefs, i.e.,
each evaluation point is in \(\mathbb R^d\), and
each coefficient is a point in \(\mathbb R^m\).
The N evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.
The K coefficients coefs are given as a gsMatrix of size K x m. Each row of coefs represents one coefficient in \(\mathbb R^m\).
The gsMatrix result contains the following data:
For every column of u, the corresponding column in the matrix result contains the gradients of the m components of the function above each other. Hence, the size of result is (d*m) x N.
Example 1:
Let \(f(s,t)\) be a bivariate scalar function, \(f:\mathbb R^2 \to \mathbb R\) (i.e., d=2, m=1), and let the evaluation point \( u_i\) be represented by the i-th column of u.
Then, result has the form
\[ \left( \begin{array}{cccc} \partial_s f(u_1) & \partial_s f(u_2) & \ldots & \partial_t f(u_{N}) \\ \partial_t f(u_1) & \partial_t f(u_2) & \ldots & \partial_t f(u_{N}) \end{array} \right) \]
Example 2:
Let \(f(s,t) = ( f_1(s,t), f_2(s,t), f_3(s,t) )\) represent a surface in space, \(f:\mathbb R^2 \to \mathbb R^3\) (i.e., d=2, m=3), and let the evaluation point \( u_i\) be represented by the i-th column of u.
Then, result has the form
\[ \left( \begin{array}{ccccccc} \partial_s f_1(u_1) & \partial_s f_1(u_2) & \ldots & \partial_s f_1(u_N) \\ \partial_t f_1(u_1) & \partial_t f_1(u_2) & \ldots & \partial_t f_1(u_N) \\ \partial_s f_2(u_1) & \partial_s f_2(u_2) & \ldots & \partial_s f_2(u_N) \\ \partial_t f_2(u_1) & \partial_t f_2(u_2) & \ldots & \partial_t f_2(u_N) \\ \partial_s f_3(u_1) & \partial_s f_3(u_2) & \ldots & \partial_s f_3(u_N) \\ \partial_t f_3(u_1) & \partial_t f_3(u_2) & \ldots & \partial_t f_3(u_N) \\ \end{array} \right) \]
[in] | u | Evaluation points as d x N-matrix. |
[in] | coefs | Coefficient matrix describing the geometry in this basis as K x m-matrix. K should equal the size() of the basis, i.e., the number basis functions. |
[in,out] | result | gsMatrix of size d*m x N, see above for format. |
where
d is the dimension of the parameter domain
m is the dimension of the physical domain
N is the number of evaluation points
K is the number of coefficients
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inlinevirtualinherited |
Evaluates the (partial) derivatives of the i-th basis function at points u into result.
See deriv_into() for detailed documentation.
Reimplemented from gsTensorBasis< d, T >.
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inlinevirtualinherited |
Return the gsDomain which represents the parameter domain of this basis. Currently unused.
Reimplemented from gsBasis< T >.
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inlinevirtualinherited |
Dimension of the (source) domain.
Implements gsFunctionSet< T >.
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inlineinherited |
Get a const-iterator to the end of the bases vector
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inlineinherited |
Get an iterator to the end of the bases vector
Evaluate the function,.
Evaluates nonzero basis functions at point u into result.
Let...
d denote the dimension of the parameter domain.
K denote the number of active (i.e., non-zero) basis functions (see active_into()). N denote the number of evaluation points.
The n evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.
The gsMatrix result contains the computed function values in the following form:
Column j of result corresponds to one evaluation point (specified by the j-th column of u). The column contains the values of all active functions "above" each other.
For example, for scalar basis functions Bi : (x,y,z)-> R, a column represents
(B1, B2, ... , BN)^T,
where the order the basis functions Bi is as returned by active() and active_into().
[in] | u | Evaluation points given as gsMatrix of size d x N. See above for details. |
[in,out] | result | gsMatrix of size K x N. See above for details. |
Reimplemented from gsTensorBasis< d, T >.
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inherited |
Evaluate all derivatives upto order n,.
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virtualinherited |
Evaluate the nonzero functions and their derivatives up to order n at points u into result.
The derivatives (the 0-th derivative is the function value) are stored in a result. result is a std::vector, where result[i] is a gsMatrix which contains the i-th derivatives.
The entries in result[0], result[1], and result[2] are ordered as in eval_into(), deriv_into(), and deriv2_into(), respectively. For i > 2, the derivatives are stored in lexicographical order, e.g. for order i = 3 and dimension 2 the derivatives are stored as follows: \( \partial_{xxx}, \, \partial_{xxy}, \, \partial_{xyy}, \, \partial_{yyy}.\, \)
[in] | u | Evaluation points, each column corresponds to one evaluation point. |
[in] | n | All derivatives up to order n are computed and stored in result. |
[in,out] | result | See above for format. |
Reimplemented from gsTensorBasis< d, T >.
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virtualinherited |
Evaluates all derivatives up to order n of the function described by coefs at points u.
Evaluation points u are given as gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.
The coefficients coefs are given as gsMatrix of size K x n, where
K is the number of (active) basis functions (=size()) and
n is the dimension of the physical domain.
Each row of coefs corresponds to the coordinates of one control point.
result is a std::vector, where the entry result[i] contains the gsMatrix corresponding to the i-th derivatives. The format of the respective entry is as in
evalFunc_into()
derivFunc_into()
deriv2Func_into()
[in] | u | |
[in] | coefs | |
[in] | n | |
[out] | result |
Evaluate the function described by coefs at points u.
This function has a default implementation that may be overridden in derived classes for higher performance.
u | evaluation points as m column vectors |
coefs | coefficient matrix describing the geometry in this basis, n columns |
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inlinevirtualinherited |
Evaluate the function described by coefs at points u, i.e., evaluates a linear combination of coefs x BasisFunctions, into result.
This function has a default implementation that may be overridden in derived classes for higher performance.
u | evaluation points as N column vectors | |
coefs | coefficient matrix describing the geometry in this basis, n columns | |
[out] | result | a matrix of size n x N with one function value as a column vector per evaluation point |
Reimplemented from gsBasis< T >.
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inherited |
Returns the i-th basis function as a gsFunction.
Note that the gsBasisFun object only holds a reference to the current basis, so it is invalidated when the basis is destroyed.
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inlineinherited |
Returns the global index of the basis function created by components of indices i,j,k (for 2d or 3d only)
Returns the global index of the basis function created by components of indices given in the vector v
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inherited |
Interpolates values on a tensor-grid of points, given in tensor form (d coordinate-wise vectors). Samples vals should be ordered as the tensor-basis coefficients
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staticinherited |
Computes the linear combination coefs * values( actives )
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inlinevirtualinherited |
Clone the source of this basis in case of rational basis, same as clone() otherwise
Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.
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inlinevirtualinherited |
Computes the indices of DoFs that match on the interface bi. The interface is assumed to be a common face between this patch and other, with an offset offset. The output is two lists of indices bndThis and bndOther, with indices that match one-to-one on the boundary bi.
NOTE: bndThis will have offset but bndOther will NOT have an offset (hence offset 0)
Reimplemented from gsBasis< T >.
Number of active basis functions at an arbitrary parameter value.
Usually, this is used for getting the active functions on one element, assuming that this number doesn't change for different parameters inside the element.
Helper function for evaluation with periodic basis.
coefs | coefficients (control points, one per row) before converting the basis into periodic. |
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virtualinherited |
Refine the basis on the area defined by the matrix boxes.
boxes is a d x n-matrix (n even), where d is the dimension of the parameter domain.
n must be even, and every 2 successive columns in the matrix define a box in the parameter domain (the first column represents the coordinates of the lower corner, the second column the coordinates of the upper corner).
Example: The input of the matrix
\[ \left[\begin{array}{cccc} 0 & 0.2 & 0.8 & 1 \\ 0.4 & 0.6 & 0.2 & 0.4 \end{array} \right] \]
results in refinement of the two boxes \([0,0.2]\times[0.4,0.6]\) and \([0.8,1]\times[0.2,0.4]\).
[in] | boxes | gsMatrix of size d x n, see above for description of size and meaning. |
[in] | refExt | Extension to be applied to the refinement boxes |
Reimplemented in gsHTensorBasis< d, T >, gsTensorBSplineBasis< d, T >, and gsTensorBSplineBasis< domainDim+1, T >.
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inlineinherited |
Increases the degree without adjusting the smoothness at inner knots, except from the knot values in knots (constrained knots of initial geometry)
This type of refinement is known as k-refinement. Note that this type of refinement is ment to be performed after one (or more) h-refinement steps the parent mesh (other), otherwise this function is equivalent to p-refinement of other.
other | parent/reference mesh determining the smoothness at the inner knots. |
i | number of k-refinement steps to perform |
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inlinevirtualinherited |
Refinement function, with different sytax for different basis.
See documentation of
gsTensorBasis::refineElements()
gsHTensorBasis::refineElements()
Reimplemented from gsBasis< T >.
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virtualinherited |
Refine basis and geometry coefficients to levels.
Refines the basis as well as the coefficients. The refinement and the format of the input depend on the implementation of refineElements().
Reimplemented in gsHTensorBasis< d, T >, and gsRationalBasis< SrcT >.
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inlinevirtualinherited |
size
Reimplemented from gsFunctionSet< T >.
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inlinevirtualinherited |
Applicable for rational bases: returns the underlying "source" (non-rational) basis
Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.
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inlinevirtualinherited |
Applicable for rational bases: returns the underlying "source" (non-rational) basis
Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.
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virtualinherited |
Returns (a bounding box for) the domain of the whole basis.
Returns a dx2 matrix, containing the two diagonally extreme corners of a hypercube.
Reimplemented from gsBasis< T >.
Returns (a bounding box for) the support of the i-th basis function.
Returns a dx2 matrix, containing the two diagonally extreme corners of a hypercube.
Reimplemented from gsBasis< T >.
Returns an interval that contains the parameter values in direction dir.
Returns a 1x2 matrix, containing the two endpoints of the interval.
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inlinevirtualinherited |
Dimension of the target space.
Reimplemented in gsPatchIdField< T >, gsGeometry< T >, gsParamField< T >, gsNormalField< T >, gsMultiBasis< T >, gsMultiBasis< real_t >, gsMultiPatch< T >, gsMultiPatch< real_t >, gsFsiLoad< T >, gsMaterialMatrixEvalSingle< T, out >, gsMaterialMatrixIntegrateSingle< T, out >, gsFunctionExpr< T >, gsJacDetField< T >, gsDetFunction< T >, gsConstantFunction< T >, gsShellStressFunction< T >, gsGradientField< T >, gsPiecewiseFunction< T >, gsSquaredDistance< T >, gsAffineFunction< T >, gsPreCICEFunction< T >, gsMappedSingleSpline< d, T >, gsBasisFun< T >, gsAbsError< T >, gsGeometrySlice< T >, gsGeometryTransform< T >, gsFuncCoordinate< T >, and gsCauchyStressFunction< T >.
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inlinevirtualinherited |
Coarsen the basis uniformly by removing groups of numKnots consecutive knots, each knot removed mul times.
This function is the oposite of gsBasis::uniformRefine
The execution of
results in no overall change in "basis". However,
is not guaranteed to keep "basis" unchanged.
Reimplemented from gsTensorBasis< d, T >.
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virtualinherited |
Coarsen the basis uniformly.
The function simultainously updates the vector coefs, representing a function in the bases, such that its new version represents the same function.
This function is equivalent to
Reimplemented in gsHTensorBasis< d, T >.
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virtualinherited |
Coarsen the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones.
The function writes a sparse matrix into the variable transfer that indicates how the functions on the coarse grid are represented as linear combinations as fine grid functions
Reimplemented from gsBasis< T >.
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virtualinherited |
Refine the basis uniformly.
The function simultainously updates the vector coefs, representing a function in the bases, such that its new version represents the same function.
This function is equivalent to
Reimplemented from gsBasis< T >.
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virtualinherited |
Refine the basis uniformly.
The function writes a sparse matrix into the variable transfer that indicates how the functions on the coarse grid are represented as linear combinations as fine grid functions
Reimplemented from gsBasis< T >.